NEW HAVEN, Conn. — Already known as a refuge for people from other lands, New Haven and its mayor are seeking to extend voting rights to illegal immigrants and other noncitizens.

Mayor John DeStefano, a Democrat, introduced four years ago a first-of-its-kind program to give noncitizens, legal or not, city resident cards. Despite crackdowns elsewhere, he has forged ahead with proposals that he says encourage differences.

“We’re a place of differences,” he said. “We’re a place that sees a strength and places a value on welcoming folks from all over.”

Dozens of American cities including New York, San Francisco and Cambridge, Mass., take a hands-off approach to pursuing illegal immigrants. While advocates say they are distancing themselves from a broken immigration system, critics accuse them of flouting federal law as “sanctuary cities.”

Presidential hopeful Newt Gingrich has vowed to cut off federal funding for such cities. Texas Gov. Rick Perry pushed a bill this year that would have prohibited cities from acting as “sanctuaries” for illegal immigrants and get local law enforcement more involved in immigration enforcement. Mitt Romney has said he opposed sanctuary cities as Massachusetts governor and, as president, he would “find the right approach” to ending them if legally possible.

President Obama has resisted calls from some Republicans to crack down on sanctuary cities. As a Democratic candidate in 2007, he said the U.S. government should address the issue by providing a rational immigration system, not by withdrawing funds from such cities.

More than 70 cities and states nationwide bar police from asking community residents who have not been arrested to prove their legal status, according to the Immigration Policy Center. For many, including New Haven, the goal is to make immigrants comfortable reaching out to police for help.

Those policies, however, do not prevent state or local police from reporting foreign-born criminals to the Department of Homeland Security.

New Haven, the home of Yale University, is a port city of 125,000 residents with a history of embracing liberal politics. The city has an estimated 10,000 to 12,000 illegal immigrants, resettles some 200 refugees annually and hosts roughly 2,000 noncitizens associated with Yale.

Mr. DeStefano long ago prohibited police from asking people about their immigration status. In addition to his pledge to seek legislative action allowing illegal immigrants to vote — a proposal that many, including Democratic Gov. Dan Malloy, see as unrealistic — the 10-term mayor last week also spoke out against a federal immigration-enforcement program, Secure Communities, which uses fingerprints collected in local jails to identify illegal immigrants.

Mr. DeStefano brushed off the Republican attacks as irrelevant to his city and suggested critics are using immigrants as scapegoats amid the weak economy.

“These are hard times right now in America,” he said. “Part of human nature is, when you’re angry to look for someone else to blame for your problems. I understand that.”

Some newcomers to New Haven are grateful to not be under scrutiny.

“Here the police give people a break. If you’re caught driving drunk or without a license, you’ll still go to court. But they don’t call immigration,” said Nicolas Gutierrez, a 63-year-old deli worker from Mexico who’s a permanent U.S. resident.